document.write( "Question 842974: We are to seat 10 boys and 7 girls in a circular arrangement. (no girl to sit next to a girl) if we the girls are adjacent and we must use all 10 boys. i mean at least 1 boy can be placed between any girl. \n" ); document.write( "

Algebra.Com's Answer #508430 by Edwin McCravy(20060) $\"\"$ $\"About$

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document.write( "Since it's a circular arrangement, we do not count the different\r\n" );
document.write( "7 possible rotations as different arrangements, so there are 6! \r\n" );
document.write( "ways to place the 7 girls.\r\n" );
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document.write( "Now we must place boys between each pair of girls.  There are 7\r\n" );
document.write( "spaces between the girls into which we must insert 7 \"things\".\r\n" );
document.write( "These 7 \"things\" are either single boys or groups of boys.  \r\n" );
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document.write( "These 7 \"things\" can either be\r\n" );
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document.write( "1. BB, BB, BB, B, B, B, B \r\n" );
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document.write( "That's 3 pairs of boys and 4 single boys.\r\n" );
document.write( "We can choose the first pair any of 10*9 ways, the second pair\r\n" );
document.write( "any of 8*7 ways and the third pair 6*5 ways. But, there are \r\n" );
document.write( "3! ways we could choose which is the 1st pair, the 2nd pair and\r\n" );
document.write( "the 3rd pair, so we must divide by 3! to avoid counting the same\r\n" );
document.write( "group of pairs more than once. So that's \r\n" );
document.write( "ways to make the \"7 things\", or \r\n" );
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document.write( "2. BBB, BB, B, B, B, B, B\r\n" );
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document.write( "That's 1 trio of boys, 1 pair of boys and 5 single boys.  We can\r\n" );
document.write( "choose the trio and of 10*9*8 ways and the pair any of 7*6 ways.\r\n" );
document.write( "So that's 10*9*8*7*6 ways to make the 7 things\r\n" );
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document.write( "3. BBBB, B, B, B, B, B, B\r\n" );
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document.write( "That's 4 boys together and 6 single boys\r\n" );
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document.write( "There are 10*9*8*7 ways to choose the 4 that sit together.\r\n" );
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document.write( "Each of those 3 groups of 7 \"things\" can be inserted in those 7 spaces\r\n" );
document.write( "in 7! ways.\r\n" );
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document.write( "My answer = 6!×[+10×9×8×7×6+10×9×8×7]×7! = 2.19469824×10¹¹\r\n" );
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document.write( "Edwin

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